There is radiation with bone scan, usually, if we are talking of subsequent bone scans, they need to be 6 months apart.
This paper
/www.ncbi.nlm.nih.gov/pubmed/27290607 indeed confirms that PET is more sensitive and specific than the bone scan. There, the patients received a bone scan and PET within 3 months. From the conclusion: "Our results indicate that BS in patients who have received PSMA PET for staging only rarely offers additional information; however, prospective studies, including a standardized integrated x-ray computed tomography (SPECT/CT) protocol, should be performed in order to confirm the presented results."
I know this does not answer your dilemma, but perhaps you should consult with a radiation oncologist if you are worried about
the radiation dose. Sometimes, even MRI can be used to put all the puzzle pieces together, with MRI there is no radiation. But, you should know that no scanning method is 100% specific.